Googology Wiki
Googology Wiki

The N-growing hierarchy is a hierarchy/notation based on the fast-growing hierarchy, created by Japanese googologist Aeton (2013) [1].


  • \([m]_0(n) = m\times n\)
  • \([m]_{\alpha+1}(n) = [m]^{n-1}_\alpha(m)\), and if \(n=1\), \([m]_{\alpha+1}(1)=[m]^0_\alpha(m)=m\)
  • \([m]_\alpha(n) = [m]_{\alpha[n]}(m)\), when \(\alpha\) is a limit ordinal and \(\alpha[n]\) is the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\).

And when \(m=10\), it can be called 10-growing hierarchy. And similarly, 3-growing hierarchy, 16-growing hierarchy, or Googol-growing hierarchy are also possible.

However, If \(m=n\), it is called Diagonal n-growing hierarchy and its notation changes as follows.

  • \((N_\alpha(n) = [n]_\alpha(n))\)
  • \(N_0(n) = n\times n=n^2\)
  • \(N_{\alpha+1}(n) = N^{n-1}_\alpha(n)\)
  • \(N_\alpha(n) = N_{\alpha[n]}(n)\)


This function is exactly equal to up-arrow notation, and probably array notation, but for that reason, when \(m=2\) and \(\alpha\geq\omega\), it does not grow well.

  • \([16]_4(8) = 16\uparrow^4 8\)
  • \([10]_{\omega+1}(100) = \{10,100,1,2\}=\) Corporal
  • \([3]^{64}_{\omega}(4)\) = Graham's number \(\lesssim[4]_{\omega+1}(65) = \{4,65,1,2\}\)
  • \([4]_{\omega^2+1}(64) = \{4,64,1,1,2\}<\) Fish number 1
  • \(N_\omega(3) = [3]_3(3) = 3\uparrow^3 3=\) Tritri
  • \(N_{\omega^2}(10) = \{10,10,10,10\}=\) General

Because of the reason that \([m]_{\omega^\omega}(n)=\{m,n+2(1)2\}\), this function doesn't match exactly over \(\{m,n(1)2\}\) level of BEAF, in \(\alpha\geq\omega^\omega\) level.

  • \(N_{\omega^{98}}(10) = [10]_{\omega^\omega}(98) = \{10,100 (1) 2\}=\) Goobol
  • \([10]_{\omega^\omega}([10]_{\omega^\omega}(98)-2)=\) goobolplex \(\approx[10]^2_{\omega^\omega}(98)\)


  1. n-growing hierarchy (Japanese Page)

See also

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea


By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's \(\mathcal{O}\)
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the \(\Sigma_1\)-soundness and the pointwise well-definedness of \(\textrm{TR}(T,n)\) for the case where \(T\) is the formalisation of the base theory


By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud

See also: Template:Googology in Asia

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)